arXiv subjects

Mujeeb ur Rehman

Publications and source records attributed to Mujeeb ur Rehman.

7 recordsLinked to original sources

arXiv2025

Operational Calculus for the nth-Level Prabhakar Type Fractional Derivative with Applications

This study investigates the nth-level Prabhakar fractional derivative, a generalization encompassing some well-known fractional derivatives. We establish its fundamental properties, particularly its relationship with the corresponding Prabhakar fractional integral. Furthermore, we develop Mikusinski-type operational calculus for this derivative, providing a framework for solving differential equations involving this operator. To illustrate its application, we present analytical solutions of two problems: a fractional order ordinary differential equation and the time fractional heat equation, both of which include the nth-level Prabhakar derivative.

math.AP↗

arXiv2025

Advancements in Hematology Analyzers: Next-Generation Technologies for Precision Diagnostics and Personalized Medicine

Hematology analyzers are essential diagnostic and monitoring tools for detecting blood diseases. Although contemporary analyzers produce only basic insights, they are often not as detailed as required under the personalized medicine paradigm. Next-Generation Hematology Analyzers (NGHAs) are revolutionary newcomers in the field, with significant advantages over regular hematology analyzers. They provide deeper insights into cellular morphology, function, and genetic profiles. This detailed information opens up possibilities for tailor-made diagnostic and therapeutic approaches in precision medicine. This review presents some revolutionary technologies that have changed hematology analyzers and provides an overview of their limitations, basic functions, and influence on clinical practice. It focuses on the integration of state-of-the-art technologies, such as microfluidics, advanced optics, artificial intelligence, flow cytometry, and digital imaging, empowering NGHAs to improve diagnostic accuracy, rapidly detect diseases, and support flexible, targeted therapy. Hints regarding point-of-care hematology testing are also provided to discuss its implications for transforming healthcare patterns. This review highlights the data management, standardization, regulatory, and ethical challenges associated with these technologies. A review tracking the current state-of-the-art and trends for the future is provided to show how these advancements may reconfigure hematology analyzer design and act as a stepping stone for future therapeutic reforms.

q-bio.OT↗

arXiv2020

Tempered and Hadamard-type fractional calculus with respect to functions

Many different types of fractional calculus have been defined, which may be categorised into broad classes according to their properties and behaviours. Two types that have been much studied in the literature are the Hadamard-type fractional calculus and tempered fractional calculus. This paper establishes a connection between these two definitions, writing one in terms of the other by making use of the theory of fractional calculus with respect to functions. By extending this connection in a natural way, a generalisation is developed which unifies several existing fractional operators: Riemann--Liouville, Caputo, classical Hadamard, Hadamard-type, tempered, and all of these taken with respect to functions. The fundamental calculus of these generalised operators is established, including semigroup and reciprocal properties as well as application to some example functions. Function spaces are constructed in which the new operators are defined and bounded. Finally, some formulae are derived for fractional integration by parts with these operators.

math.CA↗

arXiv2020

On Laplace transforms with respect to functions and their applications to fractional differential equations

An important class of fractional differential and integral operators is given by the theory of fractional calculus with respect to functions, sometimes called $Ψ$-fractional calculus. The operational calculus approach has proved useful for understanding and extending this topic of study. Motivated by fractional differential equations, we present an operational calculus approach for Laplace transforms with respect to functions and their relationship with fractional operators with respect to functions. This approach makes the generalised Laplace transforms much easier to analyse and to apply in practice. We prove several important properties of these generalised Laplace transforms, including an inversion formula, and apply it to solve some fractional differential equations, using the operational calculus approach for efficient solving.

math.CA↗

arXiv2020

Langevin equation with nonlocal boundary conditions involving a $ψ$--Caputo fractional operator

This paper studies Langevin equation with nonlocal boundary conditions involving a $ψ$--Caputo fractional derivatives operator. By the aide of fixed point techniques of Krasnoselskii and Banach, we derive new results on existence and uniqueness of the problem at hand. Further, the $ψ$-fractional Gronwall inequality and $ψ$--fractional integration by parts are employed to prove Ulam--Hyers and Ulam--Hyers--Rassias stability for the solutions. Examples are gifted to demonstrate the advantage of our major results. The proposed results here are more general than the existing results in the literature and obtain them as particular cases.

math.AP↗

arXiv2020

On Hilfer fractional difference operator

In this article, a new definition of fractional Hilfer difference operator is introduced. Definition based properties are developed and utilized to construct fixed point operator for fractional order Hilfer difference equations with initial condition. We acquire some conditions for existence, uniqueness, Ulam-Hyers and Ulam-Hyers-Rassias stability. Modified Gronwall's inequality is presented for discrete calculus with the delta difference operator.

math.GM↗

arXiv2019

Generalized substantial fractional operators and well-posedness of Cauchy problem

In this work we focus on substantial fractional integral and differential operators which play an important role in modeling anomalous diffusion. We introduce a new generalized substantial fractional integral. Generalizations of fractional substantial derivatives are also introduced both in Riemann-Liouville and Caputo sense. Furthermore, we analyze fundamental properties of these operators. Finally, we consider a class of generalized substantial fractional differential equations and discuss the existence, uniqueness and continuous dependence of solutions on initial data.

math.CA↗